Best approximations for the weighted combination of the Cauchy-Szegö kernel and its derivative in the mean

Abstract

In this paper, we study an extremal problem involving best approximation in the Hardy space $$$H^1$$$ on the unit disk $$$\mathbb D$$$. Specifically, we consider weighted combinations of the Cauchy-Szegö kernel and its derivative, parameterized by an inner funtion $$$\varphi$$$ and a complex number $$$\lambda$$$, and provide explicit formulas for the best approximation $$$e_{\varphi,z}(\lambda)$$$ by the subspace $$$H^1_0$$$. We also describe the extremal functions associated with this approximation. Our main result gives the form of $$$e_{\varphi,z}(\lambda)$$$ as a function of $$$\lambda$$$ and shows that, for a sufficiently large module of $$$\lambda$$$, the extremal function is linear in $$$\lambda$$$ and unique. We apply this result to establish a sharp inequality for holomorphic functions in the unit disk, leading to a new version of the Schwarz-Pick inequality.